Skip to main content Accessibility help
×
Home

Classification of periodic and chaotic passive limit cycles for a compass-gait biped with gait asymmetries

  • Jae-Sung Moon (a1) and Mark W. Spong (a2)

Abstract

In this paper we study the problem of passive walking for a compass-gait biped with gait asymmetries. In particular, we identify and classify bifurcations leading to chaos caused by the gait asymmetries because of unequal leg masses. We present bifurcation diagrams showing step period versus the ratio of leg masses at various walking slopes. The cell mapping method is used to find stable limit cycles as the parameters are varied. It is found that a variety of bifurcation diagrams can be grouped into six stages that consist of three expanding and three contracting stages. The analysis of each stage shows that marginally stable limit cycles exhibit period-doubling, period-remerging, and saddle-node bifurcations. We also show qualitative changes regarding chaos, i.e., generation and extinction of chaos follow cyclic patterns in passive dynamic walking.

Copyright

Corresponding author

*Corresponding author. E-mail: mspong@utdallas.edu

References

Hide All
1.Aoi, S. and Tsuchiya, K., “Bifurcation and chaos of a simple walking model driven by a rhythmic signal,” Int. J. Non-Linear Mech. 41 (3), 438446 (2006).
2.Asano, F. and Luo, Z.-W., “Pseudo Virtual Passive Dynamic Walking and Effect of Upper Body as Counterweight,” Proceedings of IEEE/RSJ International Conference on Intelligent Robots Systems, Nice, France (2008) pp. 29342939.
3.Asano, F. and Luo, Z.-W., “On Efficiency and Optimality of Asymmetric Dynamic Bipedal Gait,” Proceedings of IEEE International Conference on Robotics Automation, Kobe, Japan (2009) pp. 19721977.
4.Bier, M. and Bountis, T. C., “Remerging feigenbaum trees in dynamical systems,” Phys. Lett. A 104 (5), 239244 (1984).
5.Ephanov, A. and Hurmuzlu, Y., “Generating pathological gait patterns via the use of robotic locomotion models,” J. Technol. Health Care 10, 135146 (2002).
6.Feigenbaum, M. J., “Quantitative universality for a class of nonlinear transformations,” J. Stat. Phys. 19 (1), 2552 (1978).
7.Garcia, M., Chatterjee, A., Ruina, A. and Coleman, M., “The simplest walking model: Stability, complexity and scaling,” ASME J. Biomech. Eng. 120, 281288 (1998).
8.Goswami, A., Espiau, B. and Keramane, A., “Limit Cycles and Their Stability in a Passive Bipedal Gait,” Proceedings of IEEE International Conference on Robotics Automation, Vol. 1, Minneapolis, MN (1996) pp. 246251.
9.Goswami, A., Thuilot, B. and Espiau, B., “Compass-Like Biped Robot. Part I: Stability and Bifurcation of Passive Gaits,” INRIA Technical Report no. 2996 (INRIA, Grenoble, France. 1996).
10.Goswami, A., Thuilot, B. and Espiau, B., “A study of the passive gait of a compass-like biped robot: Symmetry and chaos,” Int. J. Robot. Res. 17 (12), 12821301 (1998).
11.Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1986).
12.Harata, Y., Asano, F., Taji, K. and Uno, Y., “Efficient Parametric Excitation Walking with Delayed Feedback Control,” Proceedings of IEEE/RSJ International Conference on Intelligent Robots Systems, St. Louis, MO (2009) pp. 29342939.
13.Hilborn, R. C., Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers (Oxford University Press, New York, 2000).
14.Howell, G. W. and Baillieul, J., “Simple Controllable Walking Mechanisms which Exhibit Bifurcations,” Proceedings of IEEE Conference on Decision Control, Tampa, FL (1998) pp. 30273032.
15.Hsu, C. S., Cell-to-Cell Mapping : A Method of Global Analysis for Nonlinear Systems (Springer-Verlag, New York, 1987).
16.Hurmuzlu, Y. and Moskowitz, G., “The role of impact in the stability of bipedal locomotion,” Dyn. Stab. Syst. 1 (3), 217234 (1986).
17.Kurz, M. J. and Stergiou, N., “Hip actuations can be used to control bifurcations and chaos in a passive dynamic walking model,” ASME J. Biomech. Eng. 129 (2), 216222 (2007).
18.McGeer, T., “Passive dynamic walking,” Int. J. Robot. Res. 9 (2), 6282 (1990).
19.Mochon, S. and McMahon, T. A., “Ballistic walking,” J. Biomech. 13 (1), 4957 (1980).
20.Mochon, S. and McMahon, T. A., “Ballistic walking: An improved model,” Math. Biosci. 52 (3–4), 241260 (1980).
21.Moon, J.-S. and Spong, M. W., “Bifurcations and Chaos in Passive Walking of a Compass-Gait Biped with Asymmetries,” Proceedings of IEEE International Conference on Robotics Automation, Anchorage, AK, (2010) pp. 17211726.
22.Ott, E., Chaos in Dynamical Systems (Cambridge University Press, New York, 1993).
23.Ott, E., Grebogi, C. and Yorke, J. A., “Controlling chaos,” Phys. Rev. Lett. 64 (11), 11961199 (1990).
24.Parker, T. S. and Chua, L. O., Practical Numerical Algorithms for Chaotic Systems (Springer-Verlag New York, NY, 1989).
25.Shinbrot, T., Grebogi, C., Ott, E. and Yorke, J. A., “Using small perturbations to control chaos,” Nature 363, 411417 (1993).
26.Spong, M. W., Hutchinson, S. and Vidyasagar, M., Robot Modeling and Control (Wiley, Hoboken, NJ, 2006).
27.Strogatz, S. H., Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Addison-Wesley, Reading, MA, 1994).
28.Suzuki, S. and Furuta, K., “Enhancement of Stabilization for Passive Walking by Chaos Control Approach,” Proceedings of IFAC Triennial World Congress, Vol. B, Barcelona, Spain (2002) pp. 133138.

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed